Past exam of the laser science and quantum information processing course of the University of Oxford / 2018 / 6 / d / Solution

The phase averaging of a polarization Bell state gives the ensemble density matrixwith $s=1$ by continuity at ${\phi }_0=0$. This follows from averaging $e^{i\phi }$ over $[-{\phi }_0,{\phi }_0]$. An observable's expectation is linear in the density matrix, so averaging the pure-state Bell function is equivalent to evaluating the mixed-state expectation. ThereforeViolation requires $\sin {\phi }_0/{\phi }_0>\sqrt{2}-1\simeq 0.41421$. The supplied graph gives the first crossing near $2.1$ radians; solving more accurately givesThe inequality is strict at the upper endpoint. On $(0,\pi )$ the sinc function decreases to zero, so this is its only crossing in that interval. For ${\phi }_0\ge \pi$, $|\sin {\phi }_0/{\phi }_0|\le 1/\pi <\sqrt{2}-1$, excluding later violation windows. The plotted Bell signal shows the same threshold without copying the source illustration.